Maybe one day we'll find an asteroid that will bring life to Mars

(but for now, we'll just look at some simulated spectra)

Erika Hornecker and Mairead Heiger

• imports
• data
• generate synthetic spectrum
  • set it up
  • profile
  • plot it
• specify the model
  • data for stan
  • build
  • sample
    • estimates
• plot the results
  • best-fit
  • diagnostics
    • trace plots
    • autocorrelation
    • effective sample size
    • corner plot
    • energy
  • goodness of fit

Imports

Index

import numpy as np

import matplotlib.pyplot as plt

from astropy.io import ascii
from astropy.table import Table

import scipy.stats as stats

import arviz as az
import stan

import nest_asyncio

import asteroids_functions as fast

import pickle

nest_asyncio.apply()

Data

Index

linelist = ascii.read("CC_CI1_Features.csv")

linelist

Table length=9

Feature	label	name	left	right	mean
str76	str66	str13	float64	float64	float64
(?) common in CI spect.	Characteristic feature of Cl1	common_feats	0.45	0.5	0.475
Fe3+ - Fe2+ charge transfers	Fe3+ - Fe2+ charge transfers	fe32_charge	0.65	0.8	0.725
Fe2+ crystal field transitions in Fe2+- bearing phyllosilicates	Fe2+ crystal field transitions in Fe2+ bearing phyllosilicates (1)	fe2_crystal_1	0.9	0.95	0.925
magnetite	Magnetite	magnetite	0.98	1.03	1.005
Fe2+ crystal field transitions in Fe2+-bearing phyllosilicates	Fe2+ crystal field transitions in Fe2+ bearing phyllosilicates (2)	fe2_crystal_2	1.1	1.15	1.125
Serpentines (from OH)	Serpentines	serpentines	1.38	1.38	1.38
Saponites (from OH, sharp bands)	Saponites indicative of OH	oh_saponites	1.39	1.42	1.405
Saponites (from H2O, sharp bands)	Saponites indicative of H2O	h2o_saponites	1.42	1.5	1.46
Combination band dude to H-O-H bending plus O-H stretching\n(presence of H2O)	H-O-H bending plus O-H stretching	hoh_oh	1.9	1.92	1.91

Generate synthetic spectrum

Index

Properties of the spectrum

Index

M = 2.2                      # set the size of the wavelength grid
dM = 0.01                    # the step in the wavelength grid
x = np.arange(0,M,dM)        # this is the wavelength grid
k_t = 6                      # set the number of features
N = len(x)                   # the number of data points we have is related to the grid
ye = 0.01                    # error (noise)

Line profile

Index

Gaussian broadening mechanisms contribute more to the shape of asteroid and meteorite reflectance spectra than Lorentzian mechanisms. Inhomogenous effects tend to produce Gaussian shapes and asteroid surfaces are replete with inhomogeneity, so we approximate its shape as purely Gaussian:

$G(x) = \phi \exp{-\frac{(x - \mu)^2}{(2\sigma^2)}}$

features, spectrum = fast.gen_spec(linelist['mean'], x, K = k_t, noise = ye)

Plot the simulated spectrum

Index

fig, ax = plt.subplots(1, 2, figsize = (12, 5))

# plot the observed spectrum
ax[0].plot(x, spectrum, '-', lw = 2)
ax[0].legend(["Simulated observed spectrum"])

# plot the components
for i in range(k_t):
    ax[1].plot(x, features['f'][i], lw = 2, label = linelist['label'][features['which']][i])

# format
fig.legend(loc = 'upper center', bbox_to_anchor = (0.5, -0.01))
ax[0].set_xlabel(r"$\lambda$ (micron)"); ax[1].set_xlabel(r"$\lambda$ (micron)")
ax[0].set_ylabel(r"Reflectance")
ax[0].set_ylim(0,1.1)
ax[1].set_ylim(0,1.1)

plt.tight_layout()

plt.show()

Specify the model

• Model
• Data
• Build
• Sample

Index

# specify the model for stan
stan_code = """ 
functions {
    real G(data real x, real phi, real mu, real sigma){
        return -phi * exp(-(x - mu)^2./(2.*sigma^2.));
    }
}

data {
    int<lower=0> K;                     // number of features (mixture components)
    int<lower=0> N;                     // number of data points (depends on how finely you sample the spectrum)
    vector[N] yo;                       // observations (literally the values of the reflectance at lambda)
    real ye;                            // error on observation
    vector[N] x;                        // wavelength grid
    real del;                           // size of the little box around each mean
    vector[K] mut;                      // means of the known features (sorted)
    vector[K] mut_l;                    // range of the known features (left bound)
    vector[K] mut_r;                    // ^^ (right bound)
}   
   
parameters {   

    // characteristics of the features (the gaussians we want to recover)
    
    real<lower = mut_l[1] - 0.001, upper = mut_r[1] + 0.001>  common_feats;    // locations of the features 
    real<lower = mut_l[2] - 0.001, upper = mut_r[2] + 0.001>   fe32_charge;
    real<lower = mut_l[3] - 0.001, upper = mut_r[3] + 0.001> fe2_crystal_1;
    real<lower = mut_l[4] - 0.001, upper = mut_r[4] + 0.001>     magnetite;
    real<lower = mut_l[5] - 0.001, upper = mut_r[5] + 0.001> fe2_crystal_2;
    real<lower = mut_l[6] - 0.001, upper = mut_r[6] + 0.001>   serpentines;
    real<lower = mut_l[7] - 0.001, upper = mut_r[7] + 0.001>  oh_saponites;
    real<lower = mut_l[8] - 0.001, upper = mut_r[8] + 0.001> h2o_saponites;
    real<lower = mut_l[9] - 0.001, upper = mut_r[9] + 0.001>        hoh_oh;
    
    vector<lower = 0>[K] sigma;                     // width of the feature
                                
    vector<lower = -0.01, upper = 1.5>[K] phi;      // (negative) depth of the feature
                                        
    real epsilon;                                   // continuum level (offset of the entire spectrum from 0 -- like an intercept)
    
}

transformed parameters{
    // the "line" of our linear regression
    vector[K] mus = [common_feats, fe32_charge, fe2_crystal_1, magnetite, fe2_crystal_2, serpentines, oh_saponites, h2o_saponites, hoh_oh]';
    vector[K] ps;                                   // placeholder
    vector[N] yt;                                   // noiseless "true" value, yo = phi*yt + epsilon
    
    // if phi of feature k is non-zero, it's contributing to the observed reflectance, so it's worth constraining its mean, SD
    // otherwise, it doesn't, but the model will still do its best to constrain them
    // so instead we don't include them in the likelihod, and their mean, SD become dependent on the initial guess    
    
    for (n in 1:N){
        for (k in 1:K){
            if (phi[k] > 0.01){
                ps[k] = G(x[n], phi[k], mus[k], sigma[k]);       // the features that are doing things
            }                                                    
            else{
                ps[k] = 0;                                       // the features that don't actually exist                                                           
            }
        }
        yt[n] = sum(ps) + epsilon;
    }

}

model{

    // priors
    
    phi ~ normal(0.5,0.5);                          // height of peak
       
    common_feats   ~ normal(mut[1], del);  // each mean is unique and ordered
    fe32_charge    ~ normal(mut[2], del);
    fe2_crystal_1  ~ normal(mut[3], del);
    magnetite      ~ normal(mut[4], del);
    fe2_crystal_2  ~ normal(mut[5], del);
    serpentines    ~ normal(mut[6], del);
    oh_saponites   ~ normal(mut[7], del);
    h2o_saponites  ~ normal(mut[8], del);
    hoh_oh         ~ normal(mut[9], del);
    
    epsilon ~ normal(1, 1);                         // continuum level

    sigma ~ normal(0.1, 0.05);                      // sigma
    
    
    // likelihood
    
    yo ~ normal(yt, ye);
}
"""

Make a dictionary of the data for stan

• Model
• Data
• Build
• Sample

Index

data = {}
data["K"] = 9                        # we have 9 features, although several of them should be 0 here
data["N"] = N                        # and N = M/dM samples on the curve
data["yo"] = spectrum                # and these are the values of the observed reflectance
data["ye"] = ye                      # with their error
data["x"] = x                        # at these values of lambda
data["del"] = 0.1                    # this is how big the little windowbox prior should be for each mu
data["mut"] = linelist['mean']       # and this is where we think they are
data["mut_l"] = linelist['left']     # ^^
data["mut_r"] = linelist['right']    # ^^

Build the model

• Model
• Data
• Build
• Sample

Index

fit = stan.build(program_code = stan_code, data = data) # build the model

Building: found in cache, done.
Messages from stanc:
Warning in '/tmp/httpstan_j_fos5k1/model_2ci5vm66.stan', line 84, column 24: Argument
    0.05 suggests there may be parameters that are not unit scale; consider
    rescaling with a multiplier (see manual section 22.12).
Warning in '/tmp/httpstan_j_fos5k1/model_2ci5vm66.stan', line 54, column 12: A
    control flow statement depends on parameter(s): phi.
Warning: Your Stan program has a parameter phi with a lower and upper bound
    in its declaration. These hard constraints are not recommended, for two
    reasons: (a) Except when there are logical or physical constraints, it is
    very unusual for you to be sure that a parameter will fall inside a
    specified range, and (b) The infinite gradient induced by a hard
    constraint can cause difficulties for Stan's sampling algorithm. As a
    consequence, we recommend soft constraints rather than hard constraints;
    for example, instead of constraining an elasticity parameter to fall
    between 0, and 1, leave it unconstrained and give it a normal(0.5,0.5)
    prior distribution.
Warning: The parameter serpentines has no priors. This means either no prior
    is provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.
Warning: The parameter oh_saponites has no priors. This means either no prior
    is provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.
Warning: The parameter magnetite has no priors. This means either no prior is
    provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.
Warning: The parameter hoh_oh has no priors. This means either no prior is
    provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.
Warning: The parameter h2o_saponites has no priors. This means either no
    prior is provided, or the prior(s) depend on data variables. In the later
    case, this may be a false positive.
Warning: The parameter fe32_charge has no priors. This means either no prior
    is provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.
Warning: The parameter fe2_crystal_2 has no priors. This means either no
    prior is provided, or the prior(s) depend on data variables. In the later
    case, this may be a false positive.
Warning: The parameter fe2_crystal_1 has no priors. This means either no
    prior is provided, or the prior(s) depend on data variables. In the later
    case, this may be a false positive.
Warning: The parameter common_feats has no priors. This means either no prior
    is provided, or the prior(s) depend on data variables. In the later case,
    this may be a false positive.

Sample

• Model
• Data
• Build
• Sample

Index

# initial guess (give it a really good one)
z = np.zeros(9)
z[features['which']] = features['phi']
ps = z

z = np.zeros(9)
z[features['which']] = features['sigma']
gs = z

initial_guess = [{'sigma':gs+np.abs(np.random.normal(0,0.01, size = 9)), 
                  'phi':np.abs(ps+np.random.normal(0,0.1, size = 9)), 
                  'epsilon':1, 
                  'mus':np.array(linelist['mean']), 'yt':spectrum}, 
                 {'sigma':gs+np.abs(np.random.normal(0,0.01, size = 9)),
                  'phi':np.abs(ps+np.random.normal(0,0.1, size = 9)),
                  'epsilon':1,
                  'mus':np.array(linelist['mean']), 'yt':spectrum},
                 {'sigma':gs+np.abs(np.random.normal(0,0.01, size = 9)), 
                  'phi':np.abs(ps+np.random.normal(0,0.1, size = 9)), 
                  'epsilon':1, 
                  'mus':np.array(linelist['mean']), 'yt':spectrum}]

draws = fit.sample(num_chains = 3, 
                   num_samples = 1000,
                   init=initial_guess, 
                   num_warmup = 1000, delta = 0.9) 

# this was run with just 1000 + 1000 steps to make sure it ran through!
# for science runs we did 5000 + 5000

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Messages received during sampling:
  Gradient evaluation took 0.005608 seconds
  1000 transitions using 10 leapfrog steps per transition would take 56.08 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.003068 seconds
  1000 transitions using 10 leapfrog steps per transition would take 30.68 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.0091 seconds
  1000 transitions using 10 leapfrog steps per transition would take 91 seconds.
  Adjust your expectations accordingly!

Get estimates of the parameters

• Model
• Data
• Build
• Sample
  • Estimates

Index

# widths, weights, intercepts, and true values
sigma = np.median(draws['sigma'], axis = 1)
phi = np.median(draws['phi'],axis = 1)
eps = np.median(draws['epsilon'],axis = 1)
yt = np.median(draws['yt'], axis = 1)

# and all the means
common_feats  = np.median(draws["common_feats"])
fe32_charge   = np.median(draws["fe32_charge"])
fe2_crystal_1 = np.median(draws["fe2_crystal_1"])
magnetite     = np.median(draws["magnetite"])
fe2_crystal_2 = np.median(draws["fe2_crystal_2"])
serpentines   = np.median(draws["serpentines"])
oh_saponites  = np.median(draws["oh_saponites"])
h2o_saponites = np.median(draws["h2o_saponites"])
hoh_oh        = np.median(draws["hoh_oh"])

means = [common_feats,fe32_charge,
         fe2_crystal_1,magnetite,
         fe2_crystal_2,serpentines,
         oh_saponites ,h2o_saponites,
         hoh_oh]

Results

• best-fit
• diagnostics
• goodness of fit

Index

Best-fit

Index

fig, ax = plt.subplots(1, 2, figsize = (12, 5))

# plot the observed spectrum
ax[0].plot(x, spectrum, '-', lw = 2, label = 'Simulated observed spectrum')

# plot the best fit spectrum
ax[0].plot(x, yt, 'k--', lw = 2, label = 'Best-fit spectrum')

# plot the best-fit components
f = []
for i,m in zip(range(9),means):
    ax[1].plot(x, -phi[i]*fast.G(x, m, sigma[i]) + eps, 'k--', zorder = 10)
    
    # and the initial guesses
    ax[0].plot(x, -initial_guess[0]['phi'][i]*fast.G(x, initial_guess[0]['mus'][i], 
                                                  initial_guess[0]['sigma'][i]) + 1, 'k:', label = ['Initial guess' if i == 0 else '__None'][0])
    

# plot the true components
for i in range(k_t):
    ax[1].plot(x, features['f'][i], lw = 3, label = linelist['label'][features['which']][i])




# format
fig.legend(loc = 'upper center', bbox_to_anchor = (0.5, -0.05))
ax[0].set_xlabel(r"$\lambda$ (micron)"); ax[1].set_xlabel(r"$\lambda$ (micron)")
ax[0].set_ylabel(r"Reflectance")
ax[0].set_ylim(0,1.1)
ax[1].set_ylim(0,1.1)
plt.show()

Diagnostics

Does the model converge? Is it well-mixed? Are there a lot of divergent transitions? Is the posterior severely biased? All of these (and more!) have been problems in other models we tried. These are some of the diagnostics we used.

• trace plots
• autocorrelation
• effective sample size
• corner plot
• energy

Index

Trace plot

Since some of the features have $\phi = 0$, we expect their other characteristics ($\mu$, $\sigma$) to behave a little weirdly.

Index

# means

ax = az.plot_trace(draws, var_names = ['common_feats',
                                       'fe32_charge',
                                       'fe2_crystal_1',
                                       'magnetite',
                                       'fe2_crystal_2',
                                       'serpentines',
                                       'oh_saponites',
                                       'h2o_saponites',
                                       'hoh_oh',
                                       'sigma',
                                       'phi',
                                       'epsilon'])



plt.tight_layout()

plt.show()

Autocorrelation time

Index

az.plot_autocorr(draws, var_names = ['phi', 'sigma', 'epsilon'], figsize = (20,40))

plt.tight_layout()
plt.show()

Effective sample size

Index

az.plot_ess(draws, var_names = ['phi', 'sigma', 'epsilon'], figsize = (20,20))

plt.tight_layout()
plt.show()

Corner plot

Index

az.rcParams["plot.max_subplots"] = 200

az.plot_pair(draws, kind = 'kde', marginals = True, var_names = ['phi'],
            figsize = (20,20))

plt.tight_layout()
plt.show()

az.plot_pair(draws, kind = 'kde', marginals = True, var_names = ['sigma'],
            figsize = (20,20))

plt.tight_layout()
plt.show()

az.plot_pair(draws, kind = 'kde', marginals = True, var_names = ['common_feats',
                                       'fe32_charge',
                                       'fe2_crystal_1',
                                       'magnetite',
                                       'fe2_crystal_2',
                                       'serpentines',
                                       'oh_saponites',
                                       'h2o_saponites',
                                       'hoh_oh'],
            figsize = (20,20))

plt.tight_layout()
plt.show()

Energy plot

Index

az.plot_energy(draws)

plt.show()

Goodness of fit

• Model
• Diagnostics

Index

# residuals
# there's no discernable pattern in lambda to the residuals 
# and they're roughly Gaussian, which is good (esp since the error is Gaussian and we're comparing observed/fit and not true/fit)
# the difference is on the order of 10%, which is the order of the error

fig = plt.figure(figsize = (6, 3))
gs = fig.add_gridspec(1, 2,  width_ratios=(4, 1),
                      left=0.1, right=0.9, bottom=0.1, top=0.9,
                      wspace=0.05, hspace=0.05)

ax = fig.add_subplot(gs[0])
ax_histy = fig.add_subplot(gs[1], sharey=ax)

ax.axhline(0, color = 'k')
ax.plot(x, data['yo'] - yt, '.')

ax_histy.tick_params(axis="y", labelleft=False)
ax_histy.hist(np.sum(features['f'], axis = 0) - 5 - yt, orientation = 'horizontal', bins = 20)

ax.set_xlabel("$\lambda$")
ax.set_ylabel("$R_{obs} - R_{fit}$")

plt.show()

/home/heigerma/.local/lib/python3.8/site-packages/IPython/core/pylabtools.py:152: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
  fig.canvas.print_figure(bytes_io, **kw)

##### r2 score
az.r2_score(data['yo'], yt)

r2        0.995293
r2_std    0.000000
dtype: float64